Structure theorem

This theorem is the main result that gives the complete classification. We state it here in a form that is suited for the classification:

Every finite abelian group can be expressed as a product of cyclic groups of prime power order. Moreover this expression is unique up to ordering of the factors and upto isomorphism

Classification

Reduction to case of prime power order groups

The above theorem also tells us that a finite abelian group is expressible as a direct product of its Sylow subgroups, so it suffices for us to classify all abelian groups of prime power order.

Dependence on partitions of the exponent

If an Abelian group of prime power order is expressed as a direct product of cyclic groups of prime power order then the sum of the prime-base logarithm of order of all the direct factors equals . Conversely, given any partition of into nonnegative integers, say , we can form an abelian group:

The forward direction of the correspondence involves taking the -fixed points of the algebraic group. The number of isomorphism classes on both sides equals the number of unordered integer partitions of , and we also have that: